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Mirrors > Home > MPE Home > Th. List > ereldm | Structured version Visualization version GIF version |
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereldm.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ereldm.2 | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Ref | Expression |
---|---|
ereldm | ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ereldm.2 | . . . 4 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | |
2 | 1 | neeq1d 2997 | . . 3 ⊢ (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅)) |
3 | ecdmn0 8773 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) | |
4 | ecdmn0 8773 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅)) |
6 | ereldm.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
7 | erdm 8735 | . . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 8 | eleq2d 2815 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋)) |
10 | 8 | eleq2d 2815 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋)) |
11 | 5, 9, 10 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 dom cdm 5678 Er wer 8722 [cec 8723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-er 8725 df-ec 8727 |
This theorem is referenced by: erth 8775 brecop 8829 eceqoveq 8841 |
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